Existence of tangent lines to Carnot-Carath\'eodory geodesics

TL;DR

This paper proves that length-minimizing curves in Carnot-Carathéodory spaces always have a tangent line at every point, establishing a fundamental regularity property without restrictions on the space or curve.

Contribution

It provides the first regularity result for length minimizers in Carnot-Carathéodory spaces without assumptions on the space or curve, including in Carnot groups.

Findings

Length-minimizers have at least one tangent curve equal to a straight horizontal line.

This regularity result applies generally, with no restrictions on the space or the curve.

The result is novel even within Carnot groups.

Abstract

We show that length minimizing curves in Carnot-Carath\'eodory spaces possess at any point at least one tangent curve (i.e., a blow-up in the nilpotent approximation) equal to a straight horizontal line. This is the first regularity result for length minimizers that holds with no assumption on either the space (e.g., its rank, step, or analyticity) or the curve, and it is novel even in the setting of Carnot groups.